Common knowledge of Omega's reliability is not sufficient. ArisKatsaris is pointing out that in your current post, Omega is allowed to condition its offer on the outcome of the coin flip. The original discussion on counterfactual mugging specifies that this is not allowed—Omega's offer is independent of the coin flip.

You do have the information. In counterfactual mugging, Omega tells you truthfully (with its trustworthiness being common knowledge) that it would have given the same offer if the coin had landed differently.

I'm starting to think that there's a deeper-than-I-thought point about the extraction of this information from the way I structured the dialogue. If all that Omega offers is a series of bet, and the TDT agent has no information about what would have Omega done if the coin toss came out differently or even if he will see Omega again, then it's not clear to me what a TDT agent should do.

If all that Omega offers is a series of bet, and the TDT agent has no information about what would have Omega done if the coin toss came out differently or even if he will see Omega again, then it's not clear to me what a TDT agent should do.

Indeed. I think that's why it's sometimes better to imagine "Omega" as some sort of stable physical process whose complete functionality we know, instead of an as an agent with mysterious motivations.